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Deformations of gerbes on smooth manifolds 387is an isomorphism of graded Lie algebras. It follows from (7.4) that the maps yieldan isomorphism of cosimplicial graded Lie algebras W H ! G DR .J.A//:Moreover, the equation (7.8) shows that if we equip H with the differential given on.N .n/ UI N .n/ U ˝ .0n/ C .Mat.A/ .0/ ˝ J N.0/ U/ loc Œ1/ byı C .0n/ .r .0/ / ˝ Id C Id ˝r can C ad .0n/ .F .0/ /D ı C .0n/ ] .r .n/ / ˝ Id C Id ˝r can C ad .0n/ ] .F .n/ // (7.9)then becomes an isomorphism of DGLA. Consider now an automorphism exp F ofthe cosimplicial graded Lie algebra H given on H by exp .0n/ F p. Note the fact thatthis morphism preserves the cosimplicial structure follows from the relation (7.7).The following result is proved by the direct calculation; see [4], Lemma 16.Lemma 7.16.exp. Fp / ı .ı Cr p ˝ Id C Id ˝r can C ad F p / ı exp. Fp /D ı Cr p ˝ Id C Id ˝r can rF p:(7.10)Therefore the morphismexp F W H ! H (7.11)conjugates the differential given by the formula (7.9) into the differential which on H is given byConsider the mapdefined as follows:ı C .0n/ .r .0/ / ˝ Id C Id ˝r can .0n/ .r can F .0/ / : (7.12)cotr W xC .J Np U/Œ1 ! C .Mat.A/ p ˝ J Np U/Œ1 (7.13)cotr.D/.a 1 ˝ j 1 ;:::;a n ˝ j n / D a 0 :::a n D.j 1 ;:::;j n /: (7.14)The map cotr is a quasiisomorphism of DGLAs (cf. [15], section 1.5.6; see also [4]Proposition 14).Lemma 7.17. For every p the mapId ˝ cotr W N p U ˝ xC .J Np U/Œ1 ! N p U ˝ C .Mat.A/ p ˝ J Np U/Œ1 (7.15)is a quasiisomorphism of DGLA, where the source and the target are equipped with thedifferentials ı Cr can C ! and ı Cr p ˝ Id C Id ˝r can rF p respectively.
388 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganProof. It is easy to see that Id˝cotr is a morphism of graded Lie algebras, which satisfies.r p˝IdCId˝rcan /ı.Id˝cotr/ D .Id˝cotr/ır can and ıı.Id˝cotr/ D .Id˝cotr/ıı.Since the domain of .Id ˝ cotr/ is the normalized complex, in view of Lemma 7.13 wealso have rF p ı .Id ˝ cotr/ D .Id ˝ cotr/ ı !. This implies that .Id ˝ cotr/ is amorphism of DGLA.To see that this map is a quasiisomorphism, introduce filtration on N p U byF i N p U D iN p U and consider the complexes xC .J Np U/Œ1 and C .Mat.A// p ˝J Np U/Œ1 equipped with the trivial filtration. The map (7.15) is a morphism of filteredcomplexes with respect to the induced filtrations on the source and the target. Thedifferentials induced on the associated graded complexes are ı (or, more precisely,Id ˝ ı) and the induced map of the associated graded objects is Id ˝ cotr which is aquasi-isomorphism. Therefore, the map (7.15) is a quasiisomorphism as claimed.The map (7.15) therefore induces a morphism Id ˝cotr W G DR .J/ ! ! H for everyW Œn ! . These morphisms are clearly compatible with the cosimplicial structureand hence induce a quasiisomorphism of cosimplicial DGLAsId ˝ cotr W G DR .J/ ! ! Hwhere the differential in the right hand side is given by (7.12).We summarize our consideration in the following:Theorem 7.18. For a any choice of 2 Isom 0 .A 01 ˝J N1 U; J.A 01 //, r2C .A 01 /and F 0 as in (7.6) the composition ˆ;r;F WD ı exp. F / ı .Id ˝ cotr/is a quasiisomorphism of cosimplicial DGLAs.ˆ;r;F W G DR .J/ ! ! G DR .J.A// (7.16)Let W V ! U be a refinement of the cover U and let .V; A / be the induced descentdatum. We will denote the corresponding cosimplicial DGLAs by G DR .J/ ! .U/ andG DR .J/ ! .V/ respectively. Then the map NW N V ! N U induces a morphism ofcosimplicial DGLAs.N/ W G DR .J.A// ! G DR .J.A // (7.17)and.N/ W G DR .J/ ! .U/ ! G DR .J/ ! .V/: (7.18)Notice also that the choice that the choice of data , r, F 0 on N U induces thecorresponding data .N/ , .N/ r, .N/ F 0 on N V. This data allows one toconstruct using the equation (7.16) the mapˆ.N/ ;.N/ r;.N/ F W G DR .J/ ! ! G DR .J.A //: (7.19)The following proposition is an easy consequence of the description of the map ˆ,and we leave the proof to the reader.
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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202 J. CuntzWe denote by N the set
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204 J. CuntzConsider the action of
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206 J. Cuntzprime numbers p and q,
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208 J. Cuntz6 Representations as cr
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210 J. CuntzProof. The spectral pro
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212 J. CuntzThe operator s 1 plays
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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List of contributorsArthur Bartels,
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